cellular map and cells

Question

Suppose we have CW-complexes $X$ and $Y$ and a subcomplex $F$ of $Y$. Let $f \colon X \to Y$ be cellular and $x \in X$ with $f(x) \in F$. Let $e \subseteq X$ be a cell of $X$ with $x \in e$. Is it then true that $f(e) \subseteq F$?

Answer 1

No - for example, let $X=S^1$ and let $Y = S^1 \vee S^1$ with minimal CW structures. Let $F$ be a copy of $S^1 \subset Y$ (say the one on the left). Then the cellular map $f: X \to Y$ which goes around both circles once is a counterexample.